### How do I evaluate standard deviation?

• I have collected responses from 85 people on their ability to undertake certain tasks.

The responses are on a five point Likert scale:

5 = Very Good, 4 = Good, 3 = Average, 2 = Poor, 1 = Very Poor,

The mean score is 2.8 and the standard deviation is 0.54.

I understand what the mean and standard deviation stand for.

My question is: how good (or bad) is this standard deviation?

In other words, are there any guidelines that can assist in the evaluation of standard deviation.

What would it mean for the SD to be good or bad here?

It is rather difficult to get such a small SD with data like this: for a mean of 2.8, the SD has to be *at least* $\sqrt{0.2\times 0.8}=0.4$. (Even if 2.8 represents a rounded value, the SD must still exceed 0.357.) An SD of 0.54 implies that no more than two people could have answered with a 5 (with 21 2's and 62 3's) and not more than six could have answered with a 1 (with 5 2's and 74 3's). This suggests the question may provide exceptionally little information because the scale does not effectively discriminate.

@whuber excellent data forsensics! But I could also imagine that either he averaged over different questions or did something wrong in his calculations. It seems hard to imagine that people did really respond so uniformly, especially when talking about their supposed abilities.

9 years ago

Standard deviations aren't "good" or "bad". They are indicators of how spread out your data is. Sometimes, in ratings scales, we want wide spread because it indicates that our questions/ratings cover the range of the group we are rating. Other times, we want a small sd because we want everyone to be "high".

For example, if you were testing the math skills of students in a calculus course, you could get a very small sd by asking them questions of elementary arithmetic such as $3+2$. But suppose you gave a more serious placement test for calculus (that is, students who passed would go into Calculus I, those who did not would take lower level courses first). You might expect a lower sd (and a higher average) among freshman at MIT than at South Podunk State, given the same test.

So. What is the purpose of your test? Who are in the sample?

(+1) Just to add a bit to the remark "Standard deviations aren't 'good' or 'bad'" - having a predictor with large standard deviation can be "good" because, in regression, it's inversely related to the standard error of a regression coefficient estimate. On the other hand, if you're concerned with precision of a measurement, then a large standard deviation is "bad". I'm guessing the original poster's interest is closer to the former but it's not clear.